Theorem sprel 47514

Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
sprel	⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6857	. 2 ⊢ (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2		sprvalpw 47510	. . . 4 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
3	2	eleq2d 2817	. . 3 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}))
4		eqeq1 2735	. . . . . 6 ⊢ (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
5	4	2rexbidv 3197	. . . . 5 ⊢ (𝑝 = 𝑋 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
6	5	elrab 3647	. . . 4 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
7	6	simprbi 496	. . 3 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})
8	3, 7	biimtrdi 253	. 2 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
9	1, 8	mpcom 38	1 ⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Vcvv 3436  𝒫 cpw 4550  {cpr 4578  ‘cfv 6481  Pairscspr 47507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-spr 47508

This theorem is referenced by:  prssspr  47515  prsprel  47517  reupr  47552